By using Schauder’s Fixed Point Theorem, we study the existence of traveling wave fronts for reaction-diffusion systems with spatio-temporal delays. In our results, we reduce the existence of traveling wave fronts to the existence of an admissible pair of upper solution and lower solution which are much easier to construct in practice.
Cite this paper
X. Han and L. Pan, "Traveling Wavefronts on Reaction Diffusion Systems with Spatio-Temporal Delays," Applied Mathematics, Vol. 4 No. 9, 2013, pp. 1278-1286. doi: 10.4236/am.2013.49172.
 S. Ai, “Traveling Wave Fronts for Generalized Fisher Equations with Spatio-Temporal Delays,” Journal of Dif ferential Equations, Vol. 232, No. 1, 2007, pp. 104-133.
 V. Capasso and L. Maddalena, “Convergence to Equilib rium States for a Reaction-Diffusion System Modelling the Spatial Spread of a Class of Bacterial and Viral Dis ease,” Journal of Mathematical Biology, Vol. 13, No. 2, 1981, pp. 173-184. doi:10.1007/BF00275212
 K. Gopalsamy, “Stability and Oscillations in Delay Dif ference Differential Equations of Population Dynamics,” Kluwer Academic, Dordrecht, 1992.
 S. Ma, “Traveling Wavefronts for Delayed Reaction-Dif fusion Systems via a Fixed Point Theorem,” Journal of Differential Equations, Vol. 171, No. 2, 2001, pp. 294-314.
 P. Popivanov, A. Slavova and P. Zecca, “Compact Trav eling Waves and Peakon Type Solutions of Several Equa tions of Mathematical Physics and Their Cellular Neural Network Realization,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 3, 2009, pp. 1453-1465.
 K. Schaaf, “Asymptotic Behavior and Traveling Wave Solutions for Parabolic Functional Differential Equa tions,” Transactions of the American Mathematical Society, Vol. 302, 1987, pp. 587-615.
 Z. Wang, W. Li and S. Ruan, “Traveling Wave Fronts in Reaction-Diffusion Systems with Spatio-Temporal De lays,” Journal of Differential Equations, Vol. 222, No. 1, 2006, pp. 185-232. doi:10.1016/ j.jde.2005.08.010
 J. Wu and X. Zou, “Traveling Wave Fronts of Reaction Diffusion Systems with Delays,” Journal of Dynamics and Differential Equations, Vol. 13, No. 3, 2001, pp. 651-687.
 X. Zou and J. Wu, “Existence of Traveling Wavefronts in Delayed Reaction-Diffusion System via Monotone Itera tion Method,” Proceedings of the American Mathematical Society, Vol. 125, 1997, pp. 2589-2598.
 D. Xu and X. Zhao, “Bistable Waves in an Epidemic Model,” Journal of Dynamics and Differential Equations, Vol. 16, No. 3, 2004, pp. 679-707.
 X. Zhao and W. Wang, “Fisher Waves in an Epidemic Model,” Discrete and Continuous Dynamical Systems— Series B, Vol. 4, No. 4, 2004, pp. 1117-1128.
 X. Zhao and D. Xiao, “The Asymptotic Speed of Spread and Traveling Waves for a Vector Disease Model,” Journal of Dynamics and Differential Equations, Vol. 18, No. 4, 2006, pp. 1001-1019.
 W. F. Yan and R. Liu, “Existence and Critical Speed of Traveling Wave Fronts in a Modified Vector Disease Model with Distributed Delay,” Journal of Dynamical and Control Systems, Vol. 18, No. 3, 2012, pp. 355-378.
 G. Lin, W. T. Li and S. G. Ruan, “Asymptotic Stability of Monostable Wavefronts in Discrete-Time Integral Recursions,” Science China Mathematics, Vol. 53, No. 5, 2010, pp. 1185-1194.
 Z.-Q. Xu and P.-X. Weng, “Traveling Waves in Nonlocal Diffusion Systems with Delays and Partial Quasi-Mono tonicity,” Applied Mathematics—A Journal of Chinese Universities, Vol. 26, No. 4, 2011, pp. 464-482.
 S.-L. Wu, H.-Q. Zhao and S.-Y. Liu, “Asymptotic Stability of Traveling Waves for Delayed Reaction-Diffusion Equations with Crossing-Monostability,” Zeitschrift für Angewandte Mathematik und Physik (ZAMP), Vol. 62, No. 3, 2011, pp. 377-397.
 H. Y. Wang, “Spreading Speeds and Traveling Waves for Non-cooperative Reaction—Diffusion Systems,” Journal of Nonlinear Science, Vol. 21, No. 5, 2011, pp. 747-783.
 X. J. Li, “Existence of Traveling Wavefronts of Nonlocal Delayed Lattice Differential Equations,” Journal of Dynamical and Control Systems, Vol. 17, No. 3, 2011, pp. 427-449. doi:10.1007/s10883-011-9124-1